{"paper":{"title":"Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jinyi Wang, Xinyu Wang, Yidong Sun","submitted_at":"2025-06-10T03:23:53Z","abstract_excerpt":"An $r$-colored Dyck path is a Dyck path with all $\\mathbf{d}$-steps having one of $r$ colors in $[r]=\\{1, 2, \\dots, r\\}$. In this paper, we consider several statistics on the set $\\mathcal{A}_{n,0}^{(r)}$ of $r$-colored Dyck paths of length $2n$ with no two consecutive $\\mathbf{d}$-steps having the same colors. Precisely, the paper studies the statistics ``number of points\" at level $\\ell$, ``number of $\\mathbf{u}$-steps\" at level $\\ell+1$, ``number of peaks\" at level $\\ell+1$ and ``number of $\\mathbf{udu}$-steps\" on the set $\\mathcal{A}_{n,0}^{(r)}$. The counting formulas of the first three s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08407","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.08407/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}